Question

I have a practice problem that I need answered, thank you

Answer 1

Given inequality:

$6x^2-x\text{ }<\text{ 2}$

Re-arranging:

$6x^2-x\text{ - 2 }<\text{ 0}$

Factorizing the expression to the left:

$\begin{gathered} 6x^2-2x\text{ + x -2 }<\text{ 0} \\ 6x^2\text{ -4x +3x -2 }<\text{ 0} \\ (3x-2)(2x+1)\text{ }<\text{ 0} \end{gathered}$

Hence:

$\begin{gathered} 3x-2\text{ }<\text{ 0} \\ 3x\text{ }<\text{ 2} \\ x\text{ }<\text{ }\frac{2}{3} \end{gathered}$

Since their product is negative. one of the factors would be positive.

$\begin{gathered} 2x\text{ + 1 > 0} \\ 2x\text{ > -1} \\ \frac{2x}{2}\text{ >}-\text{ }\frac{1}{2} \\ x\text{ > -}\frac{1}{2} \end{gathered}$

The solution on a number line:

The solution on interval notation:

$\mleft(-\frac{1}{2},\: \frac{2}{3}\mright)$